Method and apparatus for interference cancellation and detection using precoders

ABSTRACT

A method to achieve full diversity without sacrificing bandwidth and with a linear complexity in a wireless system includes the steps of orthogonally transmitting a plurality of signals utilizing multiple antennas using a corresponding plurality of precoders in a plurality of time slots, which precoders are designed using the channel information to cancel interference among the plurality of signals while achieving a maximum possible diversity of NM with low complexity for at least two users each having N transmit antennas and one receiver with M receive antennas, separating the signals in the receiver using the orthogonality of the transmitted signals, and decoding the signals independently to provide full diversity to the at least two users.

GOVERNMENT RIGHTS

This invention was made with Government Support under grant numberW911NF-04-1-0224 of MURI/ARO. The Government has certain rights in thisinvention.

RELATED APPLICATIONS

The present application is related to U.S. Provisional PatentApplication, Ser. No. 61/333,691, filed on May 11, 2010, which isincorporated herein by reference and to which priority is claimedpursuant to 35 USC 119.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to the field of wireless digital communicationusing an apparatus and a method that achieves full diversity and lowcomplexity without sacrificing bandwidth and with a linear complexity.

2. Description of the Prior Art

Antenna diversity, also known as space diversity, is any one of severalwireless diversity schemes that use two or more antennas to improve thequality and reliability of a wireless link. Often, especially in urbanand indoor environments, there is no clear line-of-sight betweentransmitter and receiver. Instead the signal is reflected along multiplepaths before finally being received. Each of these bounces can introducephase shifts, time delays, attenuations, and distortions that candestructively interfere with one another at the aperture of thereceiving antenna. Antenna diversity is especially effective atmitigating these multipath situations. This is because multiple antennasoffer a receiver several observations of the same signal. Each antennawill experience a different interference environment. Thus, if oneantenna is experiencing a deep fade, it is likely that another has asufficient signal. Collectively such a system can provide a robust link.While this is primarily seen in receiving systems (diversity reception),the analog has also proven valuable for transmitting systems (transmitdiversity) as well.

Inherently an antenna diversity scheme requires additional hardware andintegration versus a single antenna system but due to the commonality ofthe signal paths a fair amount of circuitry can be shared. Also with themultiple signals there is a greater processing demand placed on thereceiver, which can lead to tighter design requirements. Typically,however, signal reliability is paramount and using multiple antennas isan effective way to decrease the number of drop-outs and lostconnections.

In the past, systems have been devised that have used time divisionmultiple access (TDMA), frequency division multiple access (FDMA), codedivision multiple access (CDMA) or other multiple access methods toavoid interference. The disadvantage is the waste of bandwidthresources.

Another way to avoid interference is to use antenna resources at thereceiver to cancel the interference. The disadvantage of this method isthat it reduces the diversity and/or increases the complexityexponentially.

Multi-user detection schemes with simple receiver structures have beenrecently well studied. Multiple transmit and receive antennas have beenused to increase rate and improve the reliability of wireless systems.In this disclosure, we consider a multiple-antenna multi-access scenariowhere receive antennas are utilized to cancel the interference. In theprior art multiple antennas have been used to suppress the interferencefrom other users. It has been shown that one can decode each userseparately by using a large enough number of receive antennas. Morespecifically, for J users equipped with N transmit antennas, it is knownhow to cancel the interference using N J receive antennas.

To reduce the number of required receive antennas, the prior art hasprovided an interference cancellation method for users with two-transmitantennas. The method is based on the properties of orthogonal space-timeblock codes (OSTBCs) and requires a smaller number of receive antennas,i.e. as many as the number of users. This work was extended to a highernumber of transmit antennas but only for J=2 users. The common theme ofthe prior art is the utilization of the properties of the orthogonaldesigns at the transmitter to cancel the interference at the receiver.In communications, multiple-access schemes are orthogonal when an idealreceiver can completely reject arbitrarily strong unwanted signals usingdifferent basis functions than the desired signal. One such scheme istime division multiple access (TDMA), where the orthogonal basisfunctions are non-overlapping rectangular pulses (“time slots”). Anotherscheme is orthogonal frequency-division multiplexing (OFDM), whichrefers to the use, by a single transmitter, of a set of frequencymultiplexed signals with the exact minimum frequency spacing needed tomake them orthogonal so that they do not interfere with each other.

Unfortunately, the method does not work for a general case of complexconstellations, N>2 transmit antennas, and J>2 users. In fact, such anextension using orthogonal designs is impossible. Instead, it has beensuggested that a method based on quasi-orthogonal spacetime block codes(QOSTBCs) might be used. The main complexity tradeoff between OSTBCs andQOSTBCs is the symbol-by-symbol decoding versus pairwise decoding.Therefore, by a moderate increase of decoding complexity, the prior arthas extended prior multi-user detection schemes to any constellation,any number of users, and any number of transmit antennas.

Further, it is known that for M≧J receive antennas, the diversity ofeach user is equal to NM using maximum-likelihood detection and N(M−J+1) using low-complexity array-processing schemes. Note that thecomplexity of the maximum-likelihood detection increases exponentiallyas a function of the number of antennas, the number of users, and thebandwidth efficiency (measured in bits per channel use). Therefore,usually it is not practical.

The common goal and the main characteristics of the above multi-usersystems are the small number of required receive antennas and the lowcomplexity of the array-processing decoding. A receiver does not needmore than J receive antennas and the decoding is symbol-by-symbol orpairwise using low complexity array-processing methods. One drawback,however, is that if we demand low complexity, the maximum diversity ofNM is not achievable.

BRIEF SUMMARY OF THE INVENTION

The illustrated embodiment of the invention is the first method devisedthat achieves full diversity without sacrificing bandwidth and with alinear complexity. One advantage is full diversity and low complexity.The illustrated embodiments of the invention can be used in existing andfuture wireless communication systems and networks. It can simplify thedesign of MAC layer as there is no need to avoid interference, insteadwe cancel the effects of the interference. The illustrated embodimentsof the invention are expected to be used in existing and next generationwireless communications systems. It can be adopted for any TDMA/CDMAwireless communication system and network with multiple antennas, suchthose using IEEE 802.11n, IEEE 802.16e, IEEE 802.20, 4G, and WiMaxstandards. Any company in the wireless communication industry couldemploy the illustrated embodiments of the invention. The defenseindustry can also benefit from the illustrated embodiments of theinvention.

In one embodiment we consider interference cancellation for a systemwith two users when users know each other's channels. The goal is toutilize multiple antennas to cancel the interference without sacrificingthe diversity or the complexity of the system. Before, in theliterature, it was shown how a receiver with two receive antennas cancompletely cancel the interference of two users and provide a diversityof 2 for users with two transmit antennas. In this embodiment we proposea system to achieve the maximum possible diversity of 4 with lowcomplexity. One idea is to design precoders, using the channelinformation, to make it possible for different users to transmit overorthogonal spaces. Then, using the orthogonality of the transmittedsignals, the receiver can separate them and decode the signalsindependently. We analytically prove that the system provides fulldiversity to both users. In addition, we provide simulation results thatconfirm our analytical proof.

The goal in the illustrated embodiments is to utilize the channelinformation to cancel the interference without sacrificing the diversityor the complexity of the system. We have proposed a system to achievethe maximum possible diversity of N M with low complexity for two userseach with N transmit antennas and one receiver with M receive antennas.This is the first multiuser detection scheme that achieves fulldiversity while providing a linear low complexity decoding. Usingprecoders designed using the channel information, it makes it possiblefor different users to transmit over orthogonal spaces. Then, using theorthogonality of the transmitted signals, the receiver can separate themand decode the signals independently. We have analytically proven thatthe system provides full diversity to both users. In addition, weprovide simulation results that confirm our analytical results.

Our motivation is to utilize the channel information at the usertransmitters to increase the diversity of the system while keeping thelow complexity of the decoding. In other words, unlike theabove-mentioned methods, we do not use receive antennas to cancel theinterference. Instead, we use the channel information at the transmitterto design precoders that align different groups of signals alongorthogonal directions. As a result, interference suppression is achievedwithout utilizing the receive antenna resources and therefore fulldiversity is achieved naturally.

In the illustrated embodiments for the purposes of illustration, weconsider interference cancellation for a system with two users whenusers know each other channels. The goal is to utilize multiple antennasto cancel the interference without sacrificing the diversity or thecomplexity of the system. It is well known in the art how a receiverwith two receive antennas can completely cancel the interference of twousers and provide a diversity of 2 for users with two transmit antennas.The disclosed system achieves the maximum possible diversity of 4 withlow complexity. With the disclosed designed precoders, using the channelinformation, to make it possible for different users to transmit overorthogonal spaces. Then, using the orthogonality of the transmittedsignals, the receiver can separate them and decode the signalsindependently. We analytically show that the system provides fulldiversity to both users. Simulation confirms the analytical proof.

In the illustrated embodiments we disclose a scheme for two users eachwith two transmit antennas. The illustrated embodiments achieveinterference cancellation and full diversity for each user at the sametime. We show that the example then can be extended to two users eachwith more than two transmit antennas. We also extend the results to morethan two receive antennas.

Boldface letters are used to denote matrices and vectors, super-scripts( )^(T), (.)^(H) to denote the transpose and Hermitian, respectively. Wedenote the element in the ith row and the jth column of matrix Z by Z(i,j). Also, we denote the jth column of a matrix Z by Z(j). The real andimaginary parts of a scalar z are denoted by z_(R) and z_(l),respectively.

While the apparatus and method has or will be described for the sake ofgrammatical fluidity with functional explanations, it is to be expresslyunderstood that the claims, unless expressly formulated under 35 USC112, are not to be construed as necessarily limited in any way by theconstruction of “means” or “steps” limitations, but are to be accordedthe full scope of the meaning and equivalents of the definition providedby the claims under the judicial doctrine of equivalents, and in thecase where the claims are expressly formulated under 35 USC 112 are tobe accorded full statutory equivalents under 35 USC 112. The inventioncan be better visualized by turning now to the following drawingswherein like elements are referenced by like numerals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing two users, two users orthogonallytransmitting codewords, C and S, through two precoders, A and B, bymeans of at least two antennas each to a receiver with at least twoantennas.

FIG. 2 is a diagram illustrating the quasi orthogonal space in which thesignals are transmitted.

FIG. 3 is a graph comparing the bit error rate as a function of thesignal-to-noise ratio of a wireless system devised according to theillustrated embodiments to a prior art system. In FIG. 3, we compare twousers each equipped with two transmit antennas and a receiver with tworeceive antennas. We compare our results with the prior art resultsusing quadrature phase-shift keying (QPSK) for the same configurationwithout channel information at the transmitter.

FIG. 4 is a graph comparing the bit error rate as a function of thesignal-to-noise ratio of a wireless system devised according to theillustrated embodiments to a prior art system. In FIG. 4 we compare twousers each equipped with four transmit antennas and a receiver with tworeceive antennas. We compare our results with the prior art resultsusing quasi-orthogonal space-time block code (QOSTBC) for the sameconfiguration without channel information at the transmitter.

FIG. 5 is a graph comparing the bit error rate as a function of thesignal-to-noise ratio of a wireless system devised according to theillustrated embodiments to a prior art system. In FIG. 5 we compare twousers each equipped with two transmit antennas and a receiver with twoand also with three receive antennas. We compare our results with theprior art results using QOSTBC for the same configuration withoutchannel information at the transmitter.

The invention and its various embodiments can now be better understoodby turning to the following detailed description of the preferredembodiments which are presented as illustrated examples of the inventiondefined in the claims. It is expressly understood that the invention asdefined by the claims may be broader than the illustrated embodimentsdescribed below.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In illustrated embodiments, we assume a quasi-static flat Rayleighfading channel model for the channel as shown in FIG. 1. Rayleigh fadingis a statistical model for the effect of a propagation environment on aradio signal, such as that used by wireless devices. Rayleigh fadingmodels assume that the magnitude of a signal that has passed throughsuch a transmission medium (also called a communications channel) willvary randomly, or fade, according to a Rayleigh distribution, i.e. theradial component of the sum of two uncorrelated Gaussian randomvariables. Rayleigh fading is viewed as a reasonable model fortropospheric and ionospheric signal propagation as well as the effect ofheavily built-up urban environments on radio signals. Rayleigh fading ismost applicable when there is no dominant propagation along a line ofsight between the transmitter and receiver.

The path gains are independent complex Gaussian random variables andfixed during the transmission of one block. There are two users 10, 12,who send code words 30 and 32, (C, S), each with two transmit antennas16, 18 (the two pairs of antennas shown in the figure symbolically be asingle antenna symbol each) communicating over corresponding channels Gand H with one receiver 14 with two receive antennas 20 (the pair ofantennas shown in the figure symbolically be a single antenna symbol)with two corresponding signal processors 34, 36 (SP) coupled to twocorresponding maximum likelihood (ML) decoders 26, 28.

At the first two time slots, the channel matrices for Users 1 and 2 are:

$\begin{matrix}{{H = \begin{pmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{pmatrix}},{G = \begin{pmatrix}g_{11} & g_{12} \\g_{21} & g_{22}\end{pmatrix}}} & (1)\end{matrix}$respectively, where h_(ij) and g_(ij) are independent and identicallydistributed complex numbers with mean 0 and variance 1.

At the first two time slots, Users 1 and 2 transmit Alamouti codes

$\begin{matrix}{{C = \begin{pmatrix}c_{1} & {- c_{2}^{*}} \\c_{2} & c_{1}^{*}\end{pmatrix}},{S = \begin{pmatrix}s_{1} & {- s_{2}^{*}} \\s_{2} & s_{1}^{*}\end{pmatrix}}} & (2)\end{matrix}$respectively from transmitters 30, 32 in FIG. 1. At time slots 1 and 2,the received signals are respectively denoted by

$\begin{matrix}{{y^{1} = \begin{pmatrix}y_{1}^{1} \\y_{2}^{1}\end{pmatrix}},{y^{2} = \begin{pmatrix}y_{1}^{2} \\y_{2}^{2}\end{pmatrix}}} & (3)\end{matrix}$

We assume that the transmitter and receiver know the channel informationperfectly.

Let

$\begin{matrix}{{A^{1} = \begin{pmatrix}a_{11}^{1} & a_{12}^{1} \\a_{21}^{1} & a_{22}^{1}\end{pmatrix}},{A^{2} = \begin{pmatrix}a_{11}^{2} & a_{12}^{2} \\a_{21}^{2} & a_{22}^{2}\end{pmatrix}}} & (4)\end{matrix}$denote the precoders 22 of User 1 at time slots 1 and 2, respectively.Also,

$\begin{matrix}{{B^{1} = \begin{pmatrix}b_{11}^{1} & b_{12}^{1} \\b_{21}^{1} & b_{22}^{1}\end{pmatrix}},{B^{2} = \begin{pmatrix}b_{11}^{2} & b_{12}^{2} \\b_{21}^{2} & b_{22}^{2}\end{pmatrix}}} & (5)\end{matrix}$denote the precoders 24 of User 2 at time slots 1 and 2, respectively.

Our goal is to design low-complexity precoders 22, 24 to realizeinterference cancellation and full diversity for each user. The mainidea is to design precoders 22, 24 such that the two users transmit overtwo orthogonal spaces. As a result, the decoders 26, 28 can project thereceived signals to each of the orthogonal spaces and decode theinformation of each user without any interference from the other user.Later, we prove that the resultant diversity is full for each user.

We first present the precoder design for time slot 1. Then, a similardesign strategy for time slot 2 is briefly discussed. We present ourprecoder design method through the following four steps:

Step 1: Deriving the Equivalent Channel Equations:

At time slot 1, the signal model can be written as

$\begin{matrix}{y^{1} = {{\sqrt{E_{s}}{{HA}^{1}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}} + {\sqrt{E_{s}}{{GB}^{1}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}}} + W^{1}}} & (6)\end{matrix}$

At time slot 2, we have

$\begin{matrix}{y^{2} = {{\sqrt{E_{s}}{{HA}^{2}\begin{pmatrix}{- c_{2}^{*}} \\c_{1}^{*}\end{pmatrix}}} + {\sqrt{E_{s}}{{GB}^{2}\begin{pmatrix}{- s_{2}^{*}} \\s_{1}^{*}\end{pmatrix}}} + W^{2}}} & (7)\end{matrix}$where E_(S) denotes the total transmit energy of each user and

${W^{1} = \begin{pmatrix}n_{1}^{1} \\n_{2}^{1}\end{pmatrix}},{W^{2} = \begin{pmatrix}n_{1}^{2} \\n_{2}^{2}\end{pmatrix}}$

Denote the noise at the receiver at time slots 1 and 2, respectively. Weassume that n¹ ₁, n¹ ₂, n² ₂ are independent and identically distributedcomplex Gaussian noises with mean 0 and variance 1. If we let

$\begin{matrix}\begin{matrix}{{\hat{H}}^{1} = {\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{12}^{1} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{22}^{1}\end{pmatrix} = {HA}^{1}}} \\{= \begin{pmatrix}{{h_{11}a_{11}^{1}} + {h_{12}a_{21}^{1}}} & {{h_{11}a_{12}^{1}} + {h_{12}a_{22}^{1}}} \\{{h_{21}a_{11}^{1}} + {h_{22}a_{21}^{1}}} & {{h_{21}a_{12}^{1}} + {h_{22}a_{22}^{1}}}\end{pmatrix}}\end{matrix} & (8) \\\begin{matrix}{{\hat{G}}^{1} = {\begin{pmatrix}{\hat{g}}_{11}^{1} & {\hat{g}}_{12}^{1} \\{\hat{g}}_{21}^{1} & {\hat{g}}_{22}^{1}\end{pmatrix} = {GB}^{1}}} \\{= \begin{pmatrix}{{g_{11}b_{11}^{1}} + {g_{12}b_{21}^{1}}} & {{g_{11}b_{12}^{1}} + {g_{12}b_{22}^{1}}} \\{{g_{21}b_{11}^{1}} + {g_{22}b_{21}^{1}}} & {{g_{21}b_{12}^{1}} + {g_{22}b_{22}^{1}}}\end{pmatrix}}\end{matrix} & (9) \\\begin{matrix}{{\hat{H}}^{2} = {\begin{pmatrix}{\hat{h}}_{11}^{2} & {\hat{h}}_{12}^{2} \\{\hat{h}}_{21}^{2} & {\hat{h}}_{22}^{2}\end{pmatrix} = {HA}^{2}}} \\{= \begin{pmatrix}{{h_{11}a_{11}^{2}} + {h_{12}a_{21}^{2}}} & {{h_{11}a_{12}^{2}} + {h_{12}a_{22}^{2}}} \\{{h_{21}a_{11}^{2}} + {h_{22}a_{21}^{2}}} & {{h_{21}a_{12}^{2}} + {h_{22}a_{22}^{2}}}\end{pmatrix}}\end{matrix} & (10) \\\begin{matrix}{{\hat{G}}^{2} = {\begin{pmatrix}{\hat{g}}_{11}^{2} & {\hat{g}}_{12}^{2} \\{\hat{g}}_{21}^{2} & {\hat{g}}_{22}^{2}\end{pmatrix} = {GB}^{2}}} \\{= \begin{pmatrix}{{g_{11}b_{11}^{2}} + {g_{12}b_{21}^{2}}} & {{g_{11}b_{12}^{2}} + {g_{12}b_{22}^{2}}} \\{{g_{21}b_{11}^{2}} + {g_{22}b_{21}^{2}}} & {{g_{21}b_{12}^{2}} + {g_{22}b_{22}^{2}}}\end{pmatrix}}\end{matrix} & (11)\end{matrix}$then channel equations (6) and (7) can be written as

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\y_{2}^{1}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{12}^{1} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{22}^{1}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}} + {\sqrt{E_{s}}\begin{pmatrix}{\hat{g}}_{11}^{1} & {\hat{g}}_{12}^{1} \\{\hat{g}}_{21}^{1} & {\hat{g}}_{22}^{1}\end{pmatrix}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1}^{1} \\n_{2}^{1}\end{pmatrix}}} & (12) \\{\begin{pmatrix}y_{1}^{2} \\y_{2}^{2}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{2} & {\hat{h}}_{12}^{2} \\{\hat{h}}_{21}^{2} & {\hat{h}}_{22}^{2}\end{pmatrix}\begin{pmatrix}{- c_{2}^{*}} \\c_{1}^{*}\end{pmatrix}} + {\sqrt{E_{s}}\begin{pmatrix}{\hat{g}}_{11}^{2} & {\hat{g}}_{12}^{2} \\{\hat{g}}_{21}^{2} & {\hat{g}}_{22}^{2}\end{pmatrix}\begin{pmatrix}{- s_{2}^{*}} \\s_{1}^{*}\end{pmatrix}} + \begin{pmatrix}n_{1}^{2} \\n_{2}^{2}\end{pmatrix}}} & (13)\end{matrix}$

Combining equations (12) and (13), we have

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\y_{2}^{1} \\\left( y_{1}^{2} \right)^{*} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{12}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{12}^{1} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{22}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{22}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{11}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{11}^{2} \right)^{*}} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{21}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{21}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1}^{1} \\n_{2}^{1} \\\left( n_{1}^{2} \right)^{*} \\\left( n_{2}^{2} \right)^{*}\end{pmatrix}}} & (14)\end{matrix}$

Equation (14) is the equivalent channel equation and we define

$\begin{matrix}{{\overset{\sim}{H} = \begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{12}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{12}^{1} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{22}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{22}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{11}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{11}^{2} \right)^{*}} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{21}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{21}^{2} \right)^{*}}\end{pmatrix}},{\overset{\sim}{n} = \begin{pmatrix}n_{1}^{1} \\n_{2}^{1} \\\left( n_{1}^{2} \right)^{*} \\\left( n_{2}^{2} \right)^{*}\end{pmatrix}}} & (15)\end{matrix}$Step 2: Creating the Orthogonal Structure of Signal Vectors:

We aim to align signals along several orthogonal vectors to separatethem completely. From Equation (14), we know that we have four usefulsymbols of the two users. If we can transmit them along four orthogonalvectors, it is clear that we can separate them easily at the receiver.But we know that a four dimensional complex orthogonal design does notexist. So we can utilize a quasi-orthogonal design. In other words, wecan make the subspace 1 created by the first two columns of matrix{tilde over (H)} orthogonal to the subspace 2 created by the second twocolumns of matrix {tilde over (H)} as shown in FIG. 2. Then at thereceiver 14, we can separate the signals of User 1 from the signals ofUser 2.

In order to create the quasi-orthogonal structure, first, we letA ¹(1)=A ¹(2),A ²(1)=A ²(2)  (16)B ¹(1)=B ¹(2),B ²(2)  (17)

That is

$\begin{matrix}{{\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix} = \begin{pmatrix}a_{12}^{1} \\a_{22}^{1}\end{pmatrix}},{\begin{pmatrix}a_{11}^{2} \\a_{21}^{2}\end{pmatrix} = \begin{pmatrix}a_{12}^{2} \\a_{22}^{2}\end{pmatrix}}} & (18) \\{{\begin{pmatrix}b_{11}^{1} \\b_{21}^{1}\end{pmatrix} = \begin{pmatrix}b_{12}^{1} \\b_{22}^{1}\end{pmatrix}},{\begin{pmatrix}b_{11}^{2} \\b_{21}^{2}\end{pmatrix} = \begin{pmatrix}b_{12}^{2} \\b_{22}^{2}\end{pmatrix}}} & (19)\end{matrix}$

From Equations (8), (9), (18), (19), we can derive

$\begin{matrix}{{{\begin{pmatrix}{\hat{h}}_{11}^{1} \\{\hat{h}}_{21}^{1}\end{pmatrix} = \begin{pmatrix}{\hat{h}}_{12}^{1} \\{\hat{h}}_{22}^{1}\end{pmatrix}},{\begin{pmatrix}\left( {\hat{h}}_{12}^{2} \right)^{*} \\\left( {\hat{h}}_{22}^{2} \right)^{*}\end{pmatrix} = \begin{pmatrix}\left( {\hat{h}}_{11}^{2} \right)^{*} \\\left( {\hat{h}}_{21}^{2} \right)^{*}\end{pmatrix}}}{{\begin{pmatrix}{\hat{g}}_{11}^{1} \\{\hat{g}}_{21}^{1}\end{pmatrix} = \begin{pmatrix}{\hat{g}}_{12}^{1} \\{\hat{g}}_{22}^{1}\end{pmatrix}},{\begin{pmatrix}\left( {\hat{g}}_{12}^{2} \right)^{*} \\\left( {\hat{g}}_{22}^{2} \right)^{*}\end{pmatrix} = \begin{pmatrix}\left( {\hat{g}}_{11}^{2} \right)^{*} \\\left( {\hat{g}}_{21}^{2} \right)^{*}\end{pmatrix}}}} & (20)\end{matrix}$

For simplicity, equation (14) can be written as

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\y_{2}^{1} \\\left( y_{1}^{2} \right)^{*} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{12}^{2} \right)^{*}} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \overset{\sim}{n}}} & (21)\end{matrix}$

Now, we let

$\begin{matrix}{{\begin{pmatrix}{\hat{g}}_{11}^{1} \\{\hat{g}}_{21}^{1}\end{pmatrix} = {\eta_{1}\begin{pmatrix}{- \left( {\hat{h}}_{21}^{1} \right)^{*}} \\\left( {\hat{h}}_{11}^{1} \right)^{*}\end{pmatrix}}},{\begin{pmatrix}\left( {\hat{g}}_{12}^{2} \right)^{*} \\\left( {\hat{g}}_{22}^{2} \right)^{*}\end{pmatrix} = {\eta_{2}\begin{pmatrix}{- {\hat{h}}_{22}^{2}} \\{\hat{h}}_{12}^{2}\end{pmatrix}}}} & (22)\end{matrix}$where η₁ and η₂ are parameters we will determine later. Therefore,equation (21) can be written as

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\y_{2}^{1} \\\left( y_{1}^{2} \right)^{*} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} & {- {\eta_{1}\left( {\hat{h}}_{21}^{1} \right)}^{*}} & {- {\eta_{1}\left( {\hat{h}}_{21}^{1} \right)}^{*}} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} & {- {\eta_{1}\left( {\hat{h}}_{21}^{1} \right)}^{*}} & {- {\eta_{1}\left( {\hat{h}}_{21}^{1} \right)}^{*}} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} & {{- \eta_{2}}{\hat{h}}_{22}^{2}} & {\eta_{2}{\hat{h}}_{22}^{2}} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} & {\eta_{2}{\hat{h}}_{12}^{2}} & {{- \eta_{2}}{\hat{h}}_{12}^{2}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \overset{\sim}{n}}} & (23)\end{matrix}$

Note that, four symbols are transmitted along four columns of matrix{tilde over (H)} The first two columns are orthogonal to the second twocolumns. So c₁, c₂ and s₁, s₂ are transmitted in two orthogonalsubspaces as shown in FIG. 2. In this way, we can separate them andachieve interference cancellation for each user at the receiver 14.

Step 3: Designing Low-complexity Algorithms to Calculate the Parametersin the Precoders 22, 24:

In order to get the quasi-orthogonal structure given in equation (23),equation (22) shows that we need to solve the following equations

$\begin{matrix}{{\begin{pmatrix}g_{11}^{*} & g_{12}^{*} \\g_{21}^{*} & g_{22}^{*}\end{pmatrix}\begin{pmatrix}\left( b_{11}^{1} \right)^{*} \\\left( b_{21}^{1} \right)^{*}\end{pmatrix}} = {{\eta_{1}\begin{pmatrix}{- h_{21}} & {- h_{22}} \\h_{11} & h_{12}\end{pmatrix}}\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}} & (24) \\{{\begin{pmatrix}g_{11}^{*} & g_{12}^{*} \\g_{21}^{*} & g_{22}^{*}\end{pmatrix}\begin{pmatrix}\left( b_{12}^{2} \right)^{*} \\\left( b_{22}^{2} \right)^{*}\end{pmatrix}} = {{\eta_{2}\begin{pmatrix}{- h_{21}} & {- h_{22}} \\h_{11} & h_{12}\end{pmatrix}}\begin{pmatrix}a_{12}^{2} \\a_{22}^{2}\end{pmatrix}}} & (25)\end{matrix}$with the normalization conditions of the precoders 22, 24 represented by

$\begin{matrix}{{{a_{11}^{1}}^{2} + {a_{21}^{1}}^{2}} = {{{b_{11}^{1}}^{2} + {b_{21}^{1}}^{2}} = \frac{1}{2}}} & (26) \\{{{a_{12}^{2}}^{2} + {a_{22}^{2}}^{2}} = {{{b_{12}^{2}}^{2} + {b_{22}^{2}}^{2}} = \frac{1}{2}}} & (27)\end{matrix}$where we have used equations (18) and (19). Note that equations (26) and(27) are non-linear equations, if numerical algorithms are used to solvethese equations directly, the encoding complexity will be increasedexponentially with respect to the number of users and antennas. So weneed to find a low-complexity method to determine the precoderparameters. First, we consider equations (24) and (26).

From equation (24), we have

$\begin{matrix}{\begin{pmatrix}\left( b_{11}^{1} \right)^{*} \\\left( b_{21}^{1} \right)^{*}\end{pmatrix} = {{\eta_{1}\begin{pmatrix}g_{11}^{*} & g_{12}^{*} \\g_{21}^{*} & g_{22}^{*}\end{pmatrix}}^{- 1}\begin{pmatrix}{- h_{21}} & {- h_{22}} \\h_{11} & h_{12}\end{pmatrix}\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}} & (28)\end{matrix}$

Let

$\begin{matrix}{Q = {\begin{pmatrix}g_{11}^{*} & g_{12}^{*} \\g_{21}^{*} & g_{22}^{*}\end{pmatrix}^{- 1}\begin{pmatrix}{- h_{21}} & {- h_{22}} \\h_{11} & h_{12}\end{pmatrix}}} & (29)\end{matrix}$

By equations (26) and (28), we have

$\begin{matrix}{{{b_{11}^{1}}^{2} + {b_{21}^{1}}^{2}} = {{{\eta_{1}{Q\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}}}_{F}^{2} = \frac{1}{2}}} & (30)\end{matrix}$

Now, let us consider the singular value decomposition of matrix Q, i.e.,Q=UΣV ^(H) =Udiag(λ₁,λ₂)V ^(H)  (31)where U and V are unitary matrices and Σ is a diagonal matrix withnonnegative diagonal elements {λ₁, λ₂} in decreasing order. Replacingequation (31) in equation (30) results in

$\begin{matrix}{{{\eta_{1}U\;\Sigma\; V^{H} \times \begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}}_{F}^{2} = \frac{1}{2}} & (32)\end{matrix}$

We know that multiplying by a unitary matrix does not change the norm ofa vector, so we have

$\begin{matrix}{{{\eta_{1}\Sigma\; V^{H} \times \begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}}_{F}^{2} = \frac{1}{2}} & (33)\end{matrix}$

Then defining

$\begin{matrix}{\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix} = {V^{H} \times \begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}} & (34)\end{matrix}$

and replacing it in equation (33) results in

$\begin{matrix}{{{\eta_{1}{\sum\limits^{\;}\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix}}}}_{F}^{2} = {{{\eta_{1}^{2}\lambda_{1}^{2}{x_{1}}^{2}} + {\eta_{1}^{2}\lambda_{2}^{2}{x_{2}}^{2}}} = \frac{1}{2}}} & (35)\end{matrix}$

Since V^(H) is unitary, by equations (34) and (26), we have

$\begin{matrix}{{x_{1}^{2} + x_{2}^{2}} = {{{a_{11}^{1}}^{2} + {a_{21}^{1}}^{2}} = {\frac{1}{2}.}}} & (36)\end{matrix}$

If we let x ₁=x² ₁, x ₂=x² ₂, then we can replace the nonlinearequations (24), (26) by the following two linear equations:

$\begin{matrix}{{{\lambda_{1}^{2}{\overset{\_}{x}}_{1}} + {\lambda_{2}^{2}{\overset{\_}{x}}_{2}}} = \frac{1}{2\eta_{1}^{2}}} & (37) \\{{{\overset{\_}{x}}_{1} + {\overset{\_}{x}}_{2}} = \frac{1}{2}} & (38)\end{matrix}$

In the next step, we will choose the precoder parameters satisfyingequations (37) and (38). Note that. the computational complexity ofsolving these linear equations is very low compared with that of solvingequations (24), (26).

Step 4: Choosing the Precoder Parameters:

Note that in Equations (37) and (38), the number of unknown parametersis more than the number of equations. Therefore, the solution to achieveinterference-cancellation and full diversity for each user is notunique. Different solutions may lead to different coding gains anddifferent complexity. Our emphasis in the illustrated embodiment is onlow complexity. So we do not claim that our choice has the best-codinggain. In what follows, first we choose η₁.

At the first time slot, we choose η₁=1/λ₁. Then equations (37) and (38)become

$\begin{matrix}{{{\lambda_{1}^{2}{\overset{\_}{x}}_{1}} + {\lambda_{2}^{2}{\overset{\_}{x}}_{2}}} = {\frac{1}{2}\lambda_{1}^{2}}} & (39) \\{{{\overset{\_}{x}}_{1} + {\overset{\_}{x}}_{2}} = \frac{1}{2}} & (40)\end{matrix}$

It is easy to derive x ₁=½, x ₂=0. By equation (34), we have

$\begin{matrix}{\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix} = {{V \times \begin{pmatrix}\frac{1}{\sqrt{2}} \\0\end{pmatrix}} = {\frac{1}{\sqrt{2}}{V(1)}}}} & (41)\end{matrix}$

Then, by equation (28), we have

$\begin{matrix}\begin{matrix}{\begin{pmatrix}\left( b_{11}^{1} \right)^{*} \\\left( b_{21}^{1} \right)^{*}\end{pmatrix} = {\eta_{1}{Q\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix}}}} \\{= {\frac{1}{\lambda_{1}}U{\sum\limits^{\;}{V^{H}V \times \begin{pmatrix}\frac{1}{\sqrt{2}} \\0\end{pmatrix}}}}} \\{= {\frac{1}{\sqrt{2}}{U(1)}}}\end{matrix} & (42)\end{matrix}$

Finally, by equation (20), we can determine the precoders A¹ for User 1and B¹ for User 2 completely at time slot 1 as follows

$\begin{matrix}{{A^{1} = {\frac{1}{\sqrt{2}}\left\lbrack {{V(1)},{V(1)}} \right\rbrack}},{B^{1} = {\frac{1}{\sqrt{2}}\left\lbrack {{U(1)},{U(1)}} \right\rbrack}^{*}}} & (43)\end{matrix}$

At time slot 2, we need to solve equations (25) and (27). By the samemethod used for time slot 1, we can arrive at

$\begin{matrix}{{{\lambda_{1}^{2}{\overset{\_}{x}}_{1}} + {\lambda_{2}^{2}{\overset{\_}{x}}_{2}}} = \frac{1}{2\;\eta_{2}^{2}}} & (44) \\{{{\overset{\_}{x}}_{1} + {\overset{\_}{x}}_{2}} = \frac{1}{2}} & (45)\end{matrix}$

Then we choose η₂=1/λ₂. Replacing η₂ in equations (44) and (45) resultsin

$\begin{matrix}{{{\lambda_{1}^{2}{\overset{\_}{x}}_{1}} + {\lambda_{2}^{2}{\overset{\_}{x}}_{2}}} = {\frac{1}{2}\lambda_{2}^{2}}} & (46) \\{{{\overset{\_}{x}}_{1} + {\overset{\_}{x}}_{2}} = \frac{1}{2}} & (47)\end{matrix}$

It is easy to derive x ₁=0, x ₂=½. So we have

$\begin{matrix}{{\begin{pmatrix}a_{12}^{2} \\a_{22}^{2}\end{pmatrix} = {{V \times \begin{pmatrix}0 \\\frac{1}{\sqrt{2}}\end{pmatrix}} = {\frac{1}{\sqrt{2}}{V(2)}}}}{and}} & (48) \\\begin{matrix}{\begin{pmatrix}\left( b_{12}^{2} \right)^{*} \\\left( b_{22}^{2} \right)^{*}\end{pmatrix} = {\eta_{2}{Q\begin{pmatrix}a_{12}^{2} \\a_{22}^{2}\end{pmatrix}}}} \\{= {\frac{1}{\lambda_{2}}U{\sum\limits^{\;}{V^{H}V \times \begin{pmatrix}0 \\\frac{1}{\sqrt{2}}\end{pmatrix}}}}} \\{= {\frac{1}{\sqrt{2}}{U(2)}}}\end{matrix} & (49)\end{matrix}$

Finally, by equation (20), we can determine the precoders A² for User 1and B² for User 2 completely at time slot 2 as follows

$\begin{matrix}{{A^{2} = {\frac{1}{\sqrt{2}}\left\lbrack {{V(2)},{V(2)}} \right\rbrack}},{B^{2} = {\frac{1}{\sqrt{2}}\left\lbrack {{U(2)},{U(2)}} \right\rbrack}^{*}}} & (50)\end{matrix}$

So far, we have designed the precoders for both users through the abovefour steps when the channel information is known at the transmitter.

Decoding

Consider now decoding. We start with equation (21). Note that equation(21) can also be written as

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\\left( y_{1}^{2} \right)^{*} \\y_{2}^{1} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{12}^{2} \right)^{*}} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1}^{1} \\\left( n_{1}^{2} \right)^{*} \\n_{2}^{1} \\\left( n_{2}^{2} \right)^{*}\end{pmatrix}}} & (51)\end{matrix}$and we define

$\begin{matrix}{{{\overset{\_}{H} = {\begin{pmatrix}{\overset{\_}{H}}_{1} & {\overset{\_}{G}}_{1} \\{\overset{\_}{H}}_{2} & {\overset{\_}{G}}_{2}\end{pmatrix} = \begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & \left( {\hat{g}}_{12}^{2} \right)^{*} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}}\end{pmatrix}}},{\overset{\_}{n} = \begin{pmatrix}n_{1}^{1} \\\left( n_{1}^{2} \right)^{*} \\n_{2}^{1} \\\left( n_{2}^{2} \right)^{*}\end{pmatrix}}}{where}} & (52) \\{{{\overset{\_}{H}}_{1} = \begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}}\end{pmatrix}},{{\overset{\_}{G}}_{1} = \begin{pmatrix}{\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\\left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{12}^{2} \right)^{*}}\end{pmatrix}},{{\overset{\_}{H}}_{2} = \begin{pmatrix}{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}}\end{pmatrix}},{{\overset{\_}{G}}_{2} = \begin{pmatrix}{\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}}\end{pmatrix}},} & (53)\end{matrix}$

Note that H has a quasi-orthogonal structure, i.e., the first twocolumns are orthogonal to the second two columns. If we multiply bothsides of equation (51) with H ^(†) we will have

$\begin{matrix}{{{\overset{\_}{H}}^{\dagger}\begin{pmatrix}y_{1}^{1} \\\left( y_{1}^{2} \right)^{*} \\y_{2}^{1} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix}} = {{\sqrt{E_{s}}\begin{pmatrix}{{{\overset{\_}{H}}_{1}^{\dagger}{\overset{\_}{H}}_{1}} + {{\overset{\_}{H}}_{2}^{\dagger}{\overset{\_}{H}}_{2}}} & 0 \\0 & {{{\overset{\_}{G}}_{1}^{\dagger}{\overset{\_}{G}}_{1}} + {{\overset{\_}{G}}_{2}^{\dagger}{\overset{\_}{G}}_{2}}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + {{\overset{\_}{H}}^{\dagger}\overset{\_}{n}}}} & (54)\end{matrix}$

Now we define

$\begin{matrix}{{\overset{\sim}{y} = {\begin{pmatrix}{\overset{\sim}{y}}_{1} \\{\overset{\sim}{y}}_{2}\end{pmatrix} = {{\overset{\_}{H}}^{\dagger}\begin{pmatrix}y_{1}^{1} \\\left( y_{1}^{2} \right)^{*} \\y_{2}^{1} \\\left( y_{2}^{2} \right)^{*}\end{pmatrix}}}}{where}{{{\overset{\sim}{y}}_{1} = \underset{{\_\dagger}\_}{\begin{pmatrix}{\overset{\sim}{y}\left( {1,1} \right)} \\{\overset{\sim}{y}\left( {2,1} \right)}\end{pmatrix}}},{{\overset{\sim}{y}}_{2} = {\begin{pmatrix}{\overset{\sim}{y}\left( {3,1} \right)} \\{\overset{\sim}{y}\left( {4,1} \right)}\end{pmatrix}.}}}} & (55)\end{matrix}$

Note that the noise elements of H ^(t) n are correlated with covariancematrix H ^(†) H.

We can whiten this noise vector by multiplying both sides of equation(55) by the matrix H ^(†) H ^((−1/2)) as follows

$\begin{matrix}{{\left( {{\overset{\_}{H}}^{\dagger}\overset{\_}{H}} \right)^{- \frac{1}{2}}\overset{\sim}{y}} = {{\sqrt{E_{s}}\left( {{\overset{\_}{H}}^{\dagger}\overset{\_}{H}} \right)^{\frac{1}{2}}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \hat{n}}} & (56)\end{matrix}$

where n=has uncorrelated elements ˜CN (0, 1). If we defineĤ= H ₁ ^(†) H ₁ + H ₂ ^(†) H ₂  (57)Ĝ= G ₁ ^(†) G ₁ + G ₂ ^(†) G ₂  (58)

$\begin{matrix}{{\hat{n} = \begin{pmatrix}{\hat{n}}_{1} \\{\hat{n}}_{2}\end{pmatrix}},{{\hat{n}}_{1} = \begin{pmatrix}{\hat{n}\left( {1,1} \right)} \\{\hat{n}\left( {2,1} \right)}\end{pmatrix}},{{\hat{n}}_{2} = \begin{pmatrix}{\hat{n}\left( {3,1} \right)} \\{\hat{n}\left( {4,1} \right)}\end{pmatrix}}} & (59)\end{matrix}$

Then equation (56) is equivalent to the following two equations

$\begin{matrix}{{{\hat{H}}^{- \frac{1}{2}}{\overset{\sim}{y}}_{1}} = {{\sqrt{E_{s}}{{\hat{H}}^{\frac{1}{2}}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}} + {\hat{n}}_{1}}} & (60) \\{{{\hat{G}}^{- \frac{1}{2}}{\overset{\sim}{y}}_{2}} = {{\sqrt{E_{s}}{{\hat{G}}^{\frac{1}{2}}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}}} + {\hat{n}}_{2}}} & (61)\end{matrix}$

So we can realize interference cancellation and pairwise complex symboldecoding for each user. If instead of complex symbols, we use realsymbols, we can achieve symbol-by-symbol decoding using orthogonaldesigns.

When quadrature amplitude modulation (QAM) is adopted, we show that wecan further reduce the decoding complexity as follows. Note that for 2×2complex matrix

${Z = \begin{pmatrix}\alpha & \alpha \\\beta & {- \beta}\end{pmatrix}},{{Z^{\dagger}Z} = \begin{pmatrix}{{\alpha }^{2} + {\beta }^{2}} & {{\alpha }^{2} - {\beta }^{2}} \\{{\alpha }^{2} - {\beta }^{2}} & {{\alpha }^{2} + {\beta }^{2}}\end{pmatrix}},$which is a real matrix. So matrices H and G in equations (60), (61) areall real matrices. Then equations (60), (61) are equivalent to thefollowing four equations

$\begin{matrix}{{{\hat{H}}^{- \frac{1}{2}}{real}\left\{ {\overset{\sim}{y}}_{1} \right\}} = {{\sqrt{E_{s}}{{\hat{H}}^{\frac{1}{2}}\begin{pmatrix}c_{1R} \\c_{2R}\end{pmatrix}}} + {{real}\left\{ {\hat{n}}_{1} \right\}}}} & (62) \\{{{\hat{H}}^{- \frac{1}{2}}{Imag}\left\{ {\overset{\sim}{y}}_{1} \right\}} = {{\sqrt{E_{s}}{{\hat{H}}^{\frac{1}{2}}\begin{pmatrix}c_{1I} \\c_{2I}\end{pmatrix}}} + {{Imag}\left\{ {\hat{n}}_{1} \right\}}}} & (63) \\{{{\hat{G}}^{- \frac{1}{2}}{real}\left\{ {\overset{\sim}{y}}_{2} \right\}} = {{\sqrt{E_{s}}{{\hat{G}}^{\frac{1}{2}}\begin{pmatrix}s_{1R} \\s_{2R}\end{pmatrix}}} + {{real}\left\{ {\hat{n}}_{2} \right\}}}} & (64) \\{{{\hat{G}}^{- \frac{1}{2}}{Imag}\left\{ {\overset{\sim}{y}}_{2} \right\}} = {{\sqrt{E_{s}}{{\hat{G}}^{\frac{1}{2}}\begin{pmatrix}s_{1I} \\s_{2I}\end{pmatrix}}} + {{Imag}\left\{ {\hat{n}}_{2} \right\}}}} & (65)\end{matrix}$where real {z}, Imag {z} denote the real and imaginary parts of vectorz, respectively. So we can use the maximum-likelihood method to detect(c_(1R), c_(2R)), (c_(1l), c_(2l)), (s_(1R), s_(2R)), (s_(1l), s_(2l))separately. For example, by equation (62), we can detect (c_(1R),c_(2R)) by

$\begin{matrix}{{\hat{c}}_{1R},{{\hat{c}}_{2R} = {\arg{\min\limits_{c_{1R},c_{2R}}{{{{\hat{H}}^{- \frac{1}{2}}{real}\left\{ {\overset{\sim}{y}}_{1} \right\}} - {\sqrt{E_{s}}{{\hat{H}}^{\frac{1}{2}}\begin{pmatrix}c_{1R} \\c_{2R}\end{pmatrix}}}}}_{F}^{2}}}}} & (66)\end{matrix}$

Similarly, using equations (63), (64), (65), we can detect all othercodewords.

Full Diversity

Diversity is usually defined as the exponent of the ratio of probabilityof error to the signal-to-noise-ratio (SNR) as the SNR becomes verylarge or goes to infinity. In other words, theoretically as the noise ina signal goes toward zero, the probability of error in the digitalcommunication becomes very small and approaches a limit called thediversity of the system. Mathematically, the diversity order can bedefined as

$\begin{matrix}{d = {- {\lim\limits_{p->\infty}\frac{\log\; P_{e}}{\log\;\rho}}}} & (67)\end{matrix}$where p denotes the SNR and P_(e) represents the probability of error.We first consider equation (62). Here we add a real unitary rotation Rto (c_(1R), c_(2R)). Thus, the data vector

$d = {R\begin{pmatrix}c_{1R} \\c_{2R}\end{pmatrix}}$and we define the error matrix D=d− d. By equation (62), the pairwiseerror probability (PEP) can be given by the Gaussian tail function as

$\begin{matrix}{{P\left( {\left. d\longrightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} = {Q\left( \sqrt{\frac{\rho}{8}\frac{{{{\hat{H}}^{\frac{1}{2}}{RD}}}_{F}^{2}}{4N_{0}}} \right)}} & (68)\end{matrix}$

where N₀=½ is the variance of the elements of the white noise vectorreal|{{circumflex over (n)}₁} in equation (62). Now we assume H ₁ and H₂ have the following singular value decompositionsH ₁ =U ₁Λ₁ V ₁ =U ₁diag{λ₁₁,λ₁₂ }V ₁  (69)H ₂ =U ₂Λ₂ V ₂ =U ₂diag{λ₂₁,λ₂₂ }V ₂  (70)So H ₁ ^(†) H ₁ =V ₁ ^(T)Λ₁ ² V ₁ , H ₂ ^(†) H ₂ =V ₂ ^(T)Λ₂ ² V ₂.

Since H ^(t) ₁ H ₁ and H ^(t) ₂ H ₂ are both block-circulant matrices,

$V_{1} = {V_{2} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}}$ We  let${\overset{\sim}{V}}_{1} = {{\overset{\sim}{V}}_{2} = \begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}$and

${{\overset{\sim}{\Lambda}}_{1} = {{\frac{1}{\sqrt{2}}\Lambda_{1}} = {{diag}\left\{ {{\overset{\sim}{\lambda}}_{11},{\overset{\sim}{\lambda}}_{12}} \right\}}}},{{\overset{\sim}{\Lambda}}_{2} = {{\frac{1}{\sqrt{2}}\Lambda_{2}} = {{diag}{\left\{ {{\overset{\sim}{\lambda}}_{21},{\overset{\sim}{\lambda}}_{22}} \right\}.}}}}$

Therefore, equation (68) can be written as

$\begin{matrix}{{P\left( {\left. d\longrightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} = {Q\left( \sqrt{\frac{\rho\left\lbrack {D^{T}R^{T}{{\overset{\sim}{V}}_{1}^{T}\left( {{\overset{\sim}{\Lambda}}_{1}^{2} + {\overset{\sim}{\Lambda}}_{2}^{2}} \right)}{\overset{\sim}{V}}_{1}{RD}} \right\rbrack}{16}} \right)}} & (71)\end{matrix}$

By replacingΦ={tilde over (V)} ₁ RDin equation (71), we have

$\begin{matrix}{{P\left( {\left. d\longrightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} = {Q\left( \sqrt{\frac{\rho{\sum\limits_{i = 1}^{2}{\sum\limits_{j = 1}^{2}{{\Phi\left( {j,1} \right)}^{2}{{\overset{\sim}{\lambda}}_{i,j}}^{2}}}}}{16}} \right)}} & (72)\end{matrix}$

Using the inequality Q(x)≦½ exp(−x²/2) results in

$\begin{matrix}{{P\left( {\left. d\longrightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} \leq {\frac{1}{2}{\exp\left( {- \frac{\rho{\sum\limits_{i = 1}^{2}{\sum\limits_{j = 1}^{2}{{\Phi\left( {j,1} \right)}^{2}{{\overset{\sim}{\lambda}}_{i,j}}^{2}}}}}{32}} \right)}}} & (73)\end{matrix}$

Now we evaluate the distribution of Λ_(ij). We know that

$\begin{matrix}{{{\overset{\_}{H}}_{1} = {{U_{1}\begin{pmatrix}{\overset{\_}{\lambda}}_{11} & 0 \\0 & {\overset{\_}{\lambda}}_{12}\end{pmatrix}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}},{{\overset{\_}{H}}_{2} = {{U_{2}\begin{pmatrix}{\overset{\_}{\lambda}}_{21} & 0 \\0 & {\overset{\_}{\lambda}}_{22}\end{pmatrix}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}}} & (74)\end{matrix}$

Therefore,

${\begin{pmatrix}{\overset{\_}{\lambda}}_{11} \\{\overset{\_}{\lambda}}_{12}\end{pmatrix} = {{U_{1}^{\dagger}{{\overset{\_}{H}}_{1}(1)}\mspace{14mu}{and}\mspace{14mu}\begin{pmatrix}{\overset{\_}{\lambda}}_{21} \\{\overset{\_}{\lambda}}_{22}\end{pmatrix}} = {U_{2}^{\dagger}{{{\overset{\_}{H}}_{2}(1)}\;.}}}}\mspace{11mu}$

By equations (53), (8) and (10), we know that conditioned on V, eachelement of H ₁(1) and H ₂(1) will be independent and identicallydistributed complex-Gaussian random variables with mean 0 andvariance 1. Multiplying by unitary matrices U^(†) ₁ and U^(†) ₂ does notchange the distribution. So {tilde over (λ)}₁₁, {tilde over (λ)}₁₂,{tilde over (λ)}₂₁, {tilde over (λ)}₂₂ are all independent andidentically distributed complex Gaussian random variables with mean 0and variance 1. Their magnitudes, |{tilde over (λ)}_(i,j)|², areRayleigh with the probability density functionf(|{tilde over (λ)}_(i,j)|)=2|{tilde over (λ)}_(i,j)|exp(−|{tilde over(λ)}_(i,j)|²).

Using the distribution of |{tilde over (λ)}_(i,j)|², we have

$\begin{matrix}\begin{matrix}{{P\left( {d->\overset{\_}{d}} \right)} = {E\left\lbrack {P\left( {{d->\overset{\_}{d}}❘\hat{H}} \right)} \right\rbrack}} \\{= {E_{V}\left\lbrack {{E_{\hat{H}}\left\lbrack {P\left( {{d->\overset{\_}{d}}❘\hat{H}} \right)} \right\rbrack}❘V} \right\rbrack}} \\{\leq {E_{V}\left\lbrack {{E_{\hat{H}}\left\lbrack {\frac{1}{2}{\exp\left( {- \frac{\rho{\sum\limits_{i = 1}^{2}{\sum\limits_{j = 1}^{2}{{{\Phi\left( {j,1} \right)}}^{2}{{\overset{\sim}{\lambda}}_{i,j}}^{2}}}}}{32}} \right)}} \right\rbrack}❘V} \right\rbrack}} \\{= {E_{V}\left\lbrack {\frac{1}{\prod\limits_{j = 1}^{2}\left\lbrack {1 + \left( {\rho{{{\Phi\left( {j,1} \right)}}^{2}/32}} \right)} \right\rbrack^{2}}❘V} \right\rbrack}} \\{= \frac{1}{\prod\limits_{j = 1}^{2}\left\lbrack {1 + \left( {\rho{{{\Phi\left( {j,1} \right)}}^{2}/32}} \right)} \right\rbrack^{2}}}\end{matrix} & (75)\end{matrix}$

At high SNRs, one can neglect the one in the denominator and get

$\begin{matrix}{{P\left( {\left. d\longrightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} \leq {\left( \frac{\rho}{32} \right)^{- 4}{\prod\limits_{j = 1}^{2}\;{{\Phi\;\left( {j,1} \right)}}^{- 4}}}} & (76)\end{matrix}$

By equation (67), it can be shown that the diversity is 4 if we choose aproper unitary rotation matrix R such thatΠ_(j=1) ²|Φ(j,1)|≠0.The best known rotations for QAM to maximize the minimum productdistance are provided in E. Bayer-Fluckiger, F. Oggier, and E. Viterbo,“New algebraic constructions of rotated Z″•lattice constellations forthe Rayleigh fading cbannel,” IEEE Trans. Inform. Theory. vol. 50, pp.702-714, April 2004. Similarly, we can also prove that the diversity for(c_(1l), c_(2l)), (s_(1R), s_(2R)), (s_(1l), s_(2l)) are all 4.Therefore, the illustrated embodiment can achieve full diversity foreach user. When a general constellation instead of QAM is adopted,similar techniques can be used to show that the system achieves fulldiversity using equation (60).

Consider now an extension of the illustrated embodiment to two userswith more than two transmit antennas. Assume we have two users each withN=2n transmit antennas. At the first N time slots, Users 1 and 2 sendcode words

$\begin{matrix}{{C = \begin{pmatrix}{- c_{1}} & c_{1} & \ldots & c_{1} \\c_{2} & {- c_{2}} & \ldots & c_{2} \\\vdots & \vdots & \ddots & \vdots \\c_{N} & c_{N} & \ldots & {- c_{N}}\end{pmatrix}},{S = \begin{pmatrix}{- s_{1}} & s_{1} & \ldots & s_{1} \\s_{2} & {- s_{2}} & \ldots & s_{2} \\\vdots & \vdots & \ddots & \vdots \\s_{N} & s_{N} & \ldots & {- s_{N}}\end{pmatrix}}} & (77)\end{matrix}$respectively. The received signals at time slot i, i=1, . . . , N, aredenoted by

$\begin{matrix}{y^{i} = \begin{pmatrix}y_{1}^{i} \\y_{2}^{i}\end{pmatrix}} & (78)\end{matrix}$

Within these N time slots, the channel matrices for Users 1 and 2 are

$\begin{matrix}{{H = \begin{pmatrix}h_{11} & h_{12} & \ldots & h_{1\; N} \\h_{21} & h_{22} & \ldots & h_{2\; N}\end{pmatrix}},{G = \begin{pmatrix}g_{11} & g_{12} & \ldots & g_{1\; N} \\g_{21} & g_{22} & \ldots & g_{2\; N}\end{pmatrix}}} & (79)\end{matrix}$respectively. At time slot i, i=1, . . . , N, the precoders for Users 1and 2 are

$\begin{matrix}{{A^{i} = \begin{pmatrix}a_{11}^{i} & a_{12}^{i} & \ldots & a_{1\; N}^{i} \\a_{21}^{i} & a_{22}^{i} & \ldots & a_{2\; N}^{i} \\\vdots & \vdots & \ddots & \vdots \\a_{N\; 1}^{i} & a_{N\; 2}^{i} & \ldots & a_{NN}^{i}\end{pmatrix}},{B^{i} = \begin{pmatrix}b_{11}^{i} & b_{12}^{i} & \ldots & b_{1\; N}^{i} \\b_{21}^{i} & b_{22}^{i} & \ldots & b_{2\; N}^{i} \\\vdots & \vdots & \ddots & \vdots \\b_{N\; 1}^{i} & b_{N\; 2}^{i} & \ldots & b_{NN}^{i}\end{pmatrix}}} & (80)\end{matrix}$respectively. We follow the steps disclosed above to design theprecoders.Step 1: Deriving the Equivalent Channel Equations:

At time slot i, the signal model can be written as

$\begin{matrix}\begin{matrix}{y^{i} = {{\sqrt{E_{s}}{HA}^{i}{C(i)}} + {\sqrt{E_{s}}{GB}^{i}{S(i)}} + W^{i}}} \\{= {{\sqrt{E_{s}}{\hat{H}}^{i}{C(i)}} + {\sqrt{E_{s}}{\hat{G}}^{i}{S(i)}} + W^{i}}}\end{matrix} & (81)\end{matrix}$where Ĥ^(i) and Ĝ^(i) denote the equivalent channel matrices for Users 1and 2 at time slot i, respectively. Combining channel equations at thefirst N time slots, we have

$\begin{matrix}{{(82){\begin{pmatrix}y_{1}^{1} \\y_{2}^{1} \\y_{1}^{2} \\y_{2}^{2} \\\vdots \\y_{1}^{N} \\y_{2}^{N}\end{pmatrix} = {\sqrt{E_{s}}\left( \begin{matrix}{- {{\hat{H}}^{1}(1)}} & {{\hat{H}}^{1}(2)} & \ldots & {{\hat{H}}^{1}(N)} & {- {{\hat{G}}^{1}(1)}} & {{\hat{G}}^{1}(2)} & \ldots & {{\hat{G}}^{1}(N)} \\{{\hat{H}}^{2}(1)} & {- {{\hat{H}}^{2}(2)}} & \ldots & {{\hat{H}}^{2}(N)} & {{\hat{G}}^{2}(1)} & {- {{\hat{G}}^{2}(2)}} & \ldots & {{\hat{G}}^{2}(N)} \\\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\{{\hat{H}}^{N}(1)} & {{\hat{H}}^{N}(2)} & \ldots & {{- {\hat{H}}^{N}}(N)} & {{\hat{G}}^{N}(1)} & {{\hat{G}}^{N}(2)} & \ldots & {{- {\hat{G}}^{N}}(N)}\end{matrix} \right){\quad{\begin{pmatrix}c_{1} \\c_{2} \\\vdots \\c_{N} \\s_{1} \\s_{2} \\\vdots \\s_{N}\end{pmatrix} + \begin{pmatrix}n_{1}^{1} \\n_{2}^{1} \\n_{1}^{2} \\n_{2}^{2} \\\vdots \\n_{1}^{N} \\n_{2}^{N}\end{pmatrix}}}}}}} & \;\end{matrix}$

Here we let

$\begin{matrix}{{(83){\overset{\sim}{H} = \begin{pmatrix}{- {{\hat{H}}^{1}(1)}} & {{\hat{H}}^{1}(2)} & \ldots & {{\hat{H}}^{1}(N)} & {- {{\hat{G}}^{1}(1)}} & {{\hat{G}}^{1}(2)} & \ldots & {{\hat{G}}^{1}(N)} \\{{\hat{H}}^{2}(1)} & {- {{\hat{H}}^{2}(2)}} & \ldots & {{\hat{H}}^{2}(N)} & {{\hat{G}}^{2}(1)} & {- {{\hat{G}}^{2}(2)}} & \ldots & {{\hat{G}}^{2}(N)} \\\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\{{\hat{H}}^{N}(1)} & {{\hat{H}}^{N}(2)} & \ldots & {{- {\hat{H}}^{N}}(N)} & {{\hat{G}}^{N}(1)} & {{\hat{G}}^{N}(2)} & \ldots & {{- {\hat{G}}^{N}}(N)}\end{pmatrix}}}} & \;\end{matrix}$Step 2: Creating the Orthogonal Structure of Signal Vectors:

LetA ^(i)(1)=A ^(i)(2)=A ^(i)(3)= . . . =A ^(i)(N)  (84)B ^(i)(1)=B ^(i)(2)=B ^(i)(3)= . . . =B ^(i)(N)  (85)

Equations (84) and (85) will result inĤ ^(i)(1)=Ĥ ^(i)(2)= . . . =Ĥ ^(i)(N)  (86)Ĝ ^(i)(1)=Ĝ ^(i)(2)= . . . =Ĝ ^(i)(N)  (87)respectively. In order to make the symbols of Users 1 and 2 transmittedin two orthogonal subspaces, i.e., the first N columns of {tilde over(H)} are orthogonal to the second N columns of {tilde over (H)}, we let

$\begin{matrix}{\begin{pmatrix}{{\hat{G}}^{i}\left( {1,1} \right)} \\{{\hat{G}}^{i}\left( {2,1} \right)}\end{pmatrix} = {\eta_{i}\begin{pmatrix}{- {{\hat{H}}^{i}\left( {2,1} \right)}} \\{{\hat{H}}^{i}\left( {1,1} \right)}\end{pmatrix}}^{*}} & (88)\end{matrix}$Step 3: Designing Low-Complexity Algorithms to Calculate the Parametersof the Precoders:

From equation (88), we have

$\begin{matrix}{{\begin{pmatrix}g_{11} & g_{12} & \ldots & {g_{1}N} \\g_{21} & g_{22} & \ldots & {g_{2}N}\end{pmatrix}^{*}\begin{pmatrix}b_{11}^{i} \\b_{21}^{i} \\\vdots \\b_{N\; 1}^{i}\end{pmatrix}^{*}} = {{\eta_{i}\begin{pmatrix}{- h_{21}} & {- h_{22}} & \ldots & {- h_{2\; N}} \\h_{11} & h_{12} & \ldots & h_{1\; N}\end{pmatrix}}\begin{pmatrix}a_{11}^{i} \\a_{21}^{i} \\\vdots \\a_{\;{N\; 1}}^{i}\end{pmatrix}}} & (89)\end{matrix}$with nonnalization equations

$\begin{matrix}{{{{a_{11}^{i}}^{2} + {a_{21}^{i}}^{2} + \ldots + {a_{N\; 1}^{i}}^{2}} = \frac{1}{N}}{{{b_{11}^{i}}^{2} + {b_{21}^{i}}^{2} + \ldots + {b_{N\; 1}^{i}}^{2}} = \frac{1}{N}}} & (90)\end{matrix}$

Note that the channel matrices in equation (89) are not square matrices.Therefore, we cannot use the reverse matrix directly as we did for theusers with two transmit antennas above. Instead, in order to simplifythe precoder design, at the first two time slots, we let all theelements in complex vectora ^(i)=(α₁₁ ^(i)α₂₁ ^(i) . . . α_(N1) ^(i))^(T) ,i=1,2  (91)be zero except for the first two elements and also let all the elementsinb ^(i)=(b ₁₁ ^(i) b ₂₁ ^(i) . . . b _(N1) ^(i))^(T) ,i=1,2  (92)be zero except for the first two elements. By the above choices fora^(i) and b^(i), Equation (89) results in

$\begin{matrix}{{\begin{pmatrix}g_{11} & g_{12} \\g_{21} & g_{22}\end{pmatrix}^{*}\begin{pmatrix}b_{11}^{i} \\b_{21}^{i}\end{pmatrix}^{*}} = {{\eta_{i}\begin{pmatrix}{- h_{21}} & {- h_{22}} \\h_{11} & h_{12}\end{pmatrix}}\begin{pmatrix}a_{11}^{i} \\a_{21}^{i}\end{pmatrix}}} & (93)\end{matrix}$which is exactly the same as equation (28). Following the stepsdisclosed above, Equations (90) and (93) result in

$\begin{matrix}{{{{\lambda_{1}^{2}{x_{1}}^{2}} + {\lambda_{2}^{2}{x_{2}}^{2}}} = {\frac{1}{N} \cdot \frac{1}{\eta^{2}}}}{{{x_{1}}^{2} + {x_{2}}^{2}} = \frac{1}{N}}} & (94)\end{matrix}$Step 4: Choosing the Precoder Parameters:

At time slot 1, we choose η=1/λ₁. It can be shown

$\begin{matrix}{{\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix} = {\frac{1}{\sqrt{N}}\begin{pmatrix}1 \\0\end{pmatrix}}},{\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix} = {\frac{1}{\sqrt{N}}{V(1)}}}} & (95)\end{matrix}$

At time slot 2, we choose η=1/λ₂, that results in

$\begin{matrix}{{\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix} = {\frac{1}{\sqrt{N}}\begin{pmatrix}0 \\1\end{pmatrix}}},{\begin{pmatrix}a_{11}^{1} \\a_{21}^{1}\end{pmatrix} = {\frac{1}{\sqrt{N}}{V(2)}}}} & (96)\end{matrix}$

where V comes from the singular value decomposition in equation (31).

At time slots 3 and 4, the precoder design procedures are nearly thesame as that of the first two time slots. The only difference is that,we let all the elements be zero except the second two elements in botha^(i) and b^(i), i=3, 4, in order to get a square matrix like that inequation (93). Then we follow the same steps to determine the precodersat time slots 3 and 4. We repeat the same process, by shifting thewindow of two nonzero elements, until all precoders at all time slotsare designed. This completes our extension to more than two transmitantennas. For the sake of brevity, we do not include the decoding andthe proof of full diversity. They are similar in nature to what wepresented earlier for users with two transmit antennas.

Consider now the extension of the illustrated embodiment to more thantwo receive antennas. So far, we have proposed a scheme for two userseach with N transmit antennas and one receiver with two receiveantennas. Now, we consider the case of M>2 receive antennas. First, notethat if M=2m and N=nM, where m, n are positive integrals, our approachused above will still work if we adjust the dimension of the transmittedsignals, the received signals, and the channel matrices.

Second, for other cases, we show that the disclosed embodiment combinedwith antenna selection can also achieve interference cancellation andfull diversity for each user. In other words, extra antennas willprovide extra diversity and the resulting diversity of the system is NM.

For the sake of simplicity, we consider two users each with two transmitantennas and one receiver with three receive antennas. The approach fora general case of N transmit and M receive antennas is similar. Ourapproach is to select two of the three receive antennas and use themethod disclosed above for the selected antennas.

Consider the selection criterion. Note that by using the embodimentdisclosed above, as shown in equation (73), the term that determinesdiversity isΣ_(i=1) ²Σ_(j=1) ²φ(j,1)²|{tilde over (λ)}_(i,j)|².We knowΦ={tilde over (V)} ₁ RD=|Φ(1,1),Φ(2,1)|^(T)where {tilde over (V)}₁ is constant and D is the error matrix. For agiven constellation, the unitary rotation matrix R is chosen optimallyand is fixed. So we can always findφ₁=min_(∀d) _(i) _(,d) _(j) |Φ(1,1)|,i≠j.andφ₂=min_(∀d) _(i) _(,d) _(j) |Φ(1,2)|,i≠j.Now we defineφ=Σ_(i=1) ²Σ_(j=1) ²|φ_(j)|²|{tilde over (λ)}_(i,j)|².

Different choice of receive antennas will lead to different {tilde over(λ)}_(i,j) and thus different φ. To pick two out of three antennas, wehave three choices. We call the scenario that receive antennas 1 and 2are chosen as Case 1, the scenario that receive antennas 2 and 3 arechosen as Case 2, and, the scenario that receive antennas 1 and 3 arechosen as Case 3. The corresponding φ for each case is givenφ_(k)=Σ_(i=1) ²Σ_(j=1) ²|φ_(j)|²|{tilde over (λ)}_(i j) ^(k)|² ,k=1,2,3.

Our selection criterion is to pick the two receive antennas of Case iwhose corresponding φ₁ is the largest among all the three cases. Inother words, if φ₁=max(φ₁, φ₂, φ₃), then we choose the two antennascorresponding to Case i. Obviously, by this method, we can achieveinterference cancellation for each user.

We first present the proof for User 1. Let us assume the channel forUser 1 is

$H = {\begin{pmatrix}h_{11} & h_{12} \\h_{21} & h_{22} \\h_{31} & h_{32}\end{pmatrix}.}$

The channels for User 1 in Cases 1, 2, 3 are

${H_{1} = \begin{pmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{pmatrix}},{H_{2} = \begin{pmatrix}h_{11} & h_{12} \\h_{31} & h_{32}\end{pmatrix}},{and}$ ${H_{3} = \begin{pmatrix}h_{21} & h_{22} \\h_{31} & h_{32}\end{pmatrix}},$respectively. Without loss of generality, let us assumei=arg max{φ₁,φ₂,φ₃}ε{1,2,3}and the two receive antennas in case i are selected. By our selectioncriterion, we know that

$\begin{matrix}{\frac{\varphi_{1} + \varphi_{2} + \varphi_{3}}{3} \leq \varphi_{i} \leq {\varphi_{1} + \varphi_{2} + \varphi_{3}}} & (97)\end{matrix}$whereφ_(i)=|φ₁|²(|{tilde over (λ)}₁₁ ^(i)|²+|φ₂|²(|{tilde over (λ)}₁₂^(i)|²+|{tilde over (λ)}₂₂|²)  98)

Now, let us defineδ₁=|φ₁|²(|{tilde over (λ)}₁₁ ¹|²+|{tilde over (λ)}₂₁ ¹|²+|{tilde over(λ)}₂₁ ²|²)+|φ₂|²(|{tilde over (λ)}₁₂ ¹|²+|{tilde over (λ)}₂₂¹|²+|{tilde over (λ)}₂₂ ²|²)  (99)δ₂=|φ₁|²(|{tilde over (λ)}₁₁ ³|²+|{tilde over (λ)}₂₁ ³|²+|{tilde over(λ)}₁₁ ²|²)+|φ₂|²+|{tilde over (λ)}₂₂ ³|²+|{tilde over (λ)}₁₂²|²)  (100)Note thatδ₁+δ₂=σ₁+σ₂+σ₃,then by equation (97), it can be shown that

$\begin{matrix}{\frac{{2 \cdot \min}\left\{ {\delta_{1},\delta_{2}} \right\}}{3} \leq \varphi_{i} \leq {{2 \cdot \max}\left\{ {\delta_{1},\delta_{2}} \right\}}} & (101)\end{matrix}$which results in

$\begin{matrix}{{P\left( {\left. d\rightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} \leq {\frac{1}{2}{\exp\left( {- \frac{\rho\left( {{{\Phi\left( {1,1} \right)}^{2}\left( {{{\overset{\sim}{\lambda}}_{11}^{i}}^{2} + {{\overset{\sim}{\lambda}}_{21}^{i}}^{2}} \right)} + {\Phi\left( {2,1} \right)}^{2} + \left( {{{\overset{\sim}{\lambda}}_{12}^{i}}^{2} + {{\overset{\sim}{\lambda}}_{22}^{i}}^{2}} \right)} \right)}{32}} \right)}} \leq {\frac{1}{2}{\exp\left( {- \frac{{\rho\varphi}_{i}}{32}} \right)}} \leq {\frac{1}{2}{\exp\left( {- \frac{{{\rho \cdot \min}\left\{ {\delta_{1},\delta_{2}} \right\}}\;}{48}} \right)}}} & (102)\end{matrix}$and therefore

$\begin{matrix}{{P\left( d\rightarrow\overset{\_}{d} \right)} = {{E\left\lbrack {P\left( {\left. d\rightarrow\overset{\_}{d} \right.❘\hat{H}} \right)} \right\rbrack} \leq {{{E\left\lbrack {\frac{1}{2}{\exp\left( {- \frac{\rho \cdot \delta_{2}}{48}} \right)}} \right\rbrack}\Pr\left\{ {\delta_{1} > \delta_{2}} \right\}} + {{E\left\lbrack {\frac{1}{2}{\exp\left( {- \frac{\rho \cdot \delta_{1}}{48}} \right)}} \right\rbrack}\Pr\left\{ {\delta_{1} < \delta_{2}} \right\}}}}} & (103)\end{matrix}$

Let V¹, V², V³ denote the unitary matrices that come from the singularvalue decomposition given by equation (31) in the three cases,respectively. Conditioned on V¹, V², V³, it can be checked that{tilde over (λ)}₁₁ ¹,{tilde over (λ)}₂₁ ¹,{tilde over (λ)}₂₁ ²,{tildeover (λ)}₁₂ ¹,{tilde over (λ)}₂₂ ¹,{tilde over (λ)}₂₂ ²are independent and identically distributed complex Gaussian randomvariables with mean 0 and variance 1. Conditioned on V¹, V², V³, it canbe checked that{tilde over (λ)}₁₁ ³,{tilde over (λ)}₂₁ ³,{tilde over (λ)}₁₁ ²,{tildeover (λ)}₁₂ ³,{tilde over (λ)}₂₂ ³,{tilde over (λ)}₁₂ ²are also independent and identically distributed complex Gaussian randomvariables with mean 0 and variance 1. Then similar to equation (73), wehave

$\begin{matrix}{{E\left\lbrack {\frac{1}{2}{\exp\left( {- \frac{\rho - \delta_{i}}{48}} \right)}} \right\rbrack} = {E_{v^{1},v^{2},v^{3}}❘{{{E\left\lbrack {\frac{1}{2}{\exp\left( {- \frac{\rho \cdot \delta_{i}}{48}} \right)}} \right\rbrack}{{V^{1},V^{2},V^{3}}}} \leq \frac{1}{\prod\limits_{j = 1}^{2}\left\lbrack {1 + \left( {p{{\phi_{j}}^{2}/48}} \right)} \right\rbrack^{3}}}}} & (104)\end{matrix}$

Substituting equation (104) in equation (103), at high SNRs, we get

$\begin{matrix}{{P\left( d\rightarrow\overset{\_}{d} \right)} \leq {\left( \frac{\rho}{48} \right)^{- 6}{\prod\limits_{j = 1}^{2}{\phi_{j}}^{- 6}}}} & (105)\end{matrix}$

As a result, the diversity d≧6. Similarly we can prove that d≦6.Therefore, d=6 and we can achieve full diversity for User 1.

Now we prove that we can also achieve full diversity for User 2. First,we use a methodology to decode symbols of User 2 that achieves fulldiversity although it may not be optimal. Similar to equation (51), whenthere are three receive antennas, the channel equations can be writtenas

$\begin{matrix}{\begin{pmatrix}y_{1}^{1} \\\left( y_{1}^{2} \right)^{*} \\y_{2}^{1} \\\left( y_{2}^{2} \right)^{*} \\y_{3}^{1} \\\left( y_{3}^{2} \right)^{*}\end{pmatrix} = {{\sqrt{E_{s}}\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} & {\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} & \left( {\hat{g}}_{12}^{2} \right)^{*} & \left( {\hat{g}}_{12}^{2} \right)^{*} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} & {\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} & \left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}} \\{\hat{h}}_{31}^{1} & {\hat{h}}_{31}^{1} & {\hat{g}}_{31}^{1} & {\hat{g}}_{31}^{1} \\\left( {\hat{h}}_{32}^{2} \right)^{*} & {- \left( {\hat{h}}_{32}^{2} \right)^{*}} & \left( {\hat{g}}_{32}^{2} \right)^{*} & {- \left( {\hat{g}}_{32}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2} \\s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1}^{1} \\\left( n_{1}^{2} \right)^{*} \\n_{2}^{1} \\\left( n_{2}^{2} \right)^{*} \\n_{3}^{1} \\\left( n_{3}^{2} \right)^{*}\end{pmatrix}}} & (106)\end{matrix}$

By the method disclosed above, we can detect the signals of User 1 withfull diversity. Here we let

$\quad\begin{pmatrix}{\hat{c}}_{1} \\{{\hat{c}}_{2}\;}\end{pmatrix}$denote the detected signals of User 1. We subtract the term of

$\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} \\{\hat{h}}_{31}^{1} & {\hat{h}}_{31}^{1} \\\left( {\hat{h}}_{32}^{2} \right)^{*} & {- \left( {\hat{h}}_{32}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}{\hat{c}}_{1} \\{{\hat{c}}_{2}\;}\end{pmatrix}$from the channel equation to remove the effect of User 1 and will have

$\begin{matrix}{{{\begin{pmatrix}y_{1}^{1} \\\left( y_{1}^{2} \right)^{*} \\y_{2}^{1} \\\left( y_{2}^{2} \right)^{*} \\y_{3}^{1} \\\left( y_{3}^{2} \right)^{*}\end{pmatrix} - {\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} \\{\hat{h}}_{31}^{1} & {\hat{h}}_{31}^{1} \\\left( {\hat{h}}_{32}^{2} \right)^{*} & {- \left( {\hat{h}}_{32}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}{\hat{c}}_{1} \\{{\hat{c}}_{2}\;}\end{pmatrix}}} = {{\begin{pmatrix}{\hat{g}}_{11}^{1} & {\hat{g}}_{11}^{1} \\\left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{12}^{2} \right)^{*}} \\{\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}} \\{\hat{g}}_{31}^{1} & {\hat{g}}_{31}^{1} \\\left( {\hat{g}}_{32}^{2} \right)^{*} & {- \left( {\hat{g}}_{32}^{2} \right)^{*}}\end{pmatrix}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{{1R}\;}^{i} \\n_{{2R}\;}^{i} \\n_{{3R}\;}^{i} \\n_{{1I}\;}^{i} \\n_{{2I}\;}^{i} \\n_{{3I}\;}^{i}\end{pmatrix} + \sigma}}\mspace{20mu}{where}\mspace{20mu}{\sigma = {\begin{pmatrix}{\hat{h}}_{11}^{1} & {\hat{h}}_{11}^{1} \\\left( {\hat{h}}_{12}^{2} \right)^{*} & {- \left( {\hat{h}}_{12}^{2} \right)^{*}} \\{\hat{h}}_{21}^{1} & {\hat{h}}_{21}^{1} \\\left( {\hat{h}}_{22}^{2} \right)^{*} & {- \left( {\hat{h}}_{22}^{2} \right)^{*}} \\{\hat{h}}_{31}^{1} & {\hat{h}}_{31}^{1} \\\left( {\hat{h}}_{32}^{2} \right)^{*} & {- \left( {\hat{h}}_{32}^{2} \right)^{*}}\end{pmatrix}\left( {\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix} - \begin{pmatrix}{\hat{c}}_{1} \\{{\hat{c}}_{2}\;}\end{pmatrix}} \right)}}} & (107)\end{matrix}$denotes the residual error. Then we can multiply both sides of theequation (107) by

$\begin{pmatrix}{\hat{g}}_{11}^{1} & {\hat{g}}_{12}^{1} \\\left( {\hat{g}}_{12}^{2} \right)^{*} & {- \left( {\hat{g}}_{12}^{1} \right)^{*}} \\{\hat{g}}_{21}^{1} & {\hat{g}}_{21}^{1} \\\left( {\hat{g}}_{22}^{2} \right)^{*} & {- \left( {\hat{g}}_{22}^{2} \right)^{*}} \\{\hat{g}}_{31}^{1} & {\hat{g}}_{31}^{1} \\\left( {\hat{g}}_{32}^{2} \right)^{*} & {- \left( {\hat{g}}_{32}^{2} \right)^{*}}\end{pmatrix}^{1}$and use the same method discussed above to detect the signals of User 2.In what follows, we show that the method provides full diversity to User2. There are two factors that result in error for User 2. The first oneis the fading in the channel of User 2 and the second one is the errorin detecting the symbols of User 1, i.e., error propagation. Let Pr(d₂→d ₂) denote the pairwise error probability for User 2, we separate thesetwo events to havePr(d ₂ → d ₂)=Pr{d ₂ → d ₂|σ=0}Pr{σ=0}+Pr{d ₂ → d ₂|σ≠0}Pr{σ≠0}=Pr{d ₂ →d ₂|σ=0}(1−Pr{σ≠0})+Pr{d ₂ → d ₂|σ=0}Pr{σ≠0}  (108)

Since Pr(d₂→ d ₂|σ≠0)≦1, we havePr(d ₂ → d ₂)≧Pr{d ₂ → d ₂|σ=0}(1−Pr{σ≠0})+Pr{σ≠0}  (109)

Note that when σ=0, we can follow the steps discussed above to detectthe signals of User 2 and by the same technique used above, we canderive

$\begin{matrix}{{{\Pr\left\{ {{\left. d_{2}\rightarrow{\overset{\_}{d}}_{2} \right.❘\sigma} = 0} \right\}} \leq {\left( \frac{p}{32} \right)^{- 6}{\prod\limits_{j = 1}^{2}\;{\phi_{j}^{\prime}}^{- 6}}}} = {T_{1}\rho^{- 6}}} & (110)\end{matrix}$where τ₁ is a constant. From equation (105), we know thatPr{σ≠0}≦τ₂ρ⁻⁶  (111)where τ₂ is a constant. Substituting equations (110) and (111) inequation (109), we getPr(d ₂ → d ₂)≧(η+η)ρ⁻⁶  (112)

Using equation (112), it can be shown that the diversity d≧6. Also wecan show that diversity d≧6. So the diversity for User 2 is 6, i.e.,full diversity. Therefore, we can achieve full diversity for both Users1 and 2 which can also be confirmed by the simulations below.

Note that when we complete the detection of the symbols of User 2, wecan remove the effects of User 2 using the detected symbols of User 2and re-detect the symbols of User 1. Simulation results show that suchan iteration improves the coding gain. Finally, a similar antennaselection method at the receiver results in a diversity of N M for ageneral case of N transmit and M receive antennas.

Simulation

Consider now simulation results that confirm our analysis stated above.We assume a quasi-static Rayleigh channel. The performance of ourproposed scheme is shown in FIGS. 3, 4 and 5. In each figure, the curvesfor Users 1 and 2 are identical. In FIG. 3, we consider two users eachequipped with two transmit antennas and a receiver with two receiveantennas. We compare our results using QPSK with the conventionalresults in J. Kazemitabar and H. Jafarkhani, “Multiuser interferencecancellation and detection for users with more than two transmitantennas:” IEEE Trans. on Communications, vol. 56, no, 4, pp. 574-583,April 2008 (hereinafter Kazemitabar) for the same configuration withoutchannel information at the transmitter. With two receive antennas, themulti-user detection (MUD) method offered in Kazemitabar can cancel theinterference and provides a diversity of 2. The method of theillustrated embodiment can also cancel the interference completely butprovides a diversity of 4 by utilizing the channel information at thetransmitter.

We also present the results for system with no interference. This is thesame system when User 2 does not exist that can be easily achieved byG=0. Simulation results confirm that we have achieved interferencecancellation completely.

Next, we present results for two users each with four transmit antennasand one receiver with two receive antennas in FIG. 4. We compare theperformance of our method with the multiuser detection method inKazemitabar using QOSTBC. As shown in FIG. 4, our scheme can achieve adiversity of 8, i.e., full diversity, by using channel information,while the MUD method using QOSTBC with no. channel information can onlyachieve a diversity of 4.

Further, we show the results for two users each with two transmitantennas and one receiver with two or three receive antennas in FIG. 5.By increasing the number of receive antennas from two to three, thediversity increases from 4 to 6. Therefore, extra receive antennas willprovide extra diversity and the resulting diversity of the system is N Mwhich confirms our theoretical analysis.

In summary, we have considered interference cancellation for a systemwith two users when users know each other's channels. The goal is toutilize the channel information to cancel the interference withoutsacrificing the diversity or the complexity of the system. We havedisclosed a system to achieve the maximum possible diversity of N M withlow complexity for two users each with N transmit antennas and onereceiver with M receive antennas. This is the first multiuser detectionscheme that achieves full diversity while providing a linear lowcomplexity decoding. The disclosed methodology is directed to designingprecoders, using the channel information, to make it possible fordifferent users to transmit over orthogonal spaces. Then, using theorthogonality of the transmitted signals, the receiver can separate themand decode the signals independently. We have analytically proved thatthe system provides full diversity to both users. In addition, weprovide simulation results that confirm our analytical results.

Many alterations and modifications may be made by those having ordinaryskill in the art without departing from the spirit and scope of theinvention. Therefore, it must be understood that the illustratedembodiment has been set forth only for the purposes of example and thatit should not be taken as limiting the invention as defined by thefollowing invention and its various embodiments.

Therefore, it must be understood that the illustrated embodiment hasbeen set forth only for the purposes of example and that it should notbe taken as limiting the invention as defined by the following claims.For example, notwithstanding the fact that the elements of a claim areset forth below in a certain combination, it must be expresslyunderstood that the invention includes other combinations of fewer, moreor different elements, which are disclosed in above even when notinitially claimed in such combinations. A teaching that two elements arecombined in a claimed combination is further to be understood as alsoallowing for a claimed combination in which the two elements are notcombined with each other, but may be used alone or combined in othercombinations. The excision of any disclosed element of the invention isexplicitly contemplated as within the scope of the invention.

The words used in this specification to describe the invention and itsvarious embodiments are to be understood not only in the sense of theircommonly defined meanings, but to include by special definition in thisspecification structure, material or acts beyond the scope of thecommonly defined meanings. Thus if an element can be understood in thecontext of this specification as including more than one meaning, thenits use in a claim must be understood as being generic to all possiblemeanings supported by the specification and by the word itself.

The definitions of the words or elements of the following claims are,therefore, defined in this specification to include not only thecombination of elements which are literally set forth, but allequivalent structure, material or acts for performing substantially thesame function in substantially the same way to obtain substantially thesame result. In this sense it is therefore contemplated that anequivalent substitution of two or more elements may be made for any oneof the elements in the claims below or that a single element may besubstituted for two or more elements in a claim. Although elements maybe described above as acting in certain combinations and even initiallyclaimed as such, it is to be expressly understood that one or moreelements from a claimed combination can in some cases be excised fromthe combination and that the claimed combination may be directed to asubcombination or variation of a subcombination.

Insubstantial changes from the claimed subject matter as viewed by aperson with ordinary skill in the art, now known or later devised, areexpressly contemplated as being equivalently within the scope of theclaims. Therefore, obvious substitutions now or later known to one withordinary skill in the art are defined to be within the scope of thedefined elements.

The claims are thus to be understood to include what is specificallyillustrated and described above, what is conceptionally equivalent, whatcan be obviously substituted and also what essentially incorporates theessential idea of the invention.

We claim:
 1. A method to achieve full diversity without sacrificingbandwidth and with a linear complexity in a wireless system comprising:orthogonally transmitting a plurality of signals utilizing multipleantennas using a corresponding plurality of precoders in a plurality oftime slots, which precoders are designed using the channel informationto cancel interference among the plurality of signals while achieving adiversity of NM with low complexity for at least two users each having Ntransmit antennas and one receiver with M receive antennas; separatingthe signals in the receiver using the orthogonality of the transmittedsignals; and decoding the signals independently to provide diversity ofNM to the at least two users; where orthogonally transmitting aplurality of signals utilizing multiple antennas using a correspondingplurality of precoders comprises: determining for each time slot oftransmission an equivalent channel equation characterized by a channelmatrix {tilde over (H)} and a noise vector ñ for the at least two users;generating an orthogonal structure of signal vectors in correspondingprecoders for the at least two users by utilizing a quasi-orthogonaldesign comprised of a subspace 1 created by the first two columns ofmatrix {tilde over (H)} orthogonal to a subspace 2 created by the secondtwo columns of matrix {tilde over (H)}.
 2. The method of claim 1 wheredecoding the signals independently to provide diversity of NM to the atleast two users comprises achieving a diversity of at least
 4. 3. Themethod of claim 1 where decoding the signals independently to providediversity of NM to the at least two users further comprises achievingdiversity of NM with low computational complexity for more than twousers in the wireless system.
 4. The method of claim 1 where decodingthe signals independently to provide diversity of NM to the at least twousers further comprises achieving diversity while providing a linear lowcomplexity decoding with any number of users.
 5. The method of claim 1where generating an orthogonal structure of signal vectors incorresponding precoders for the at least two users by utilizing aquasi-orthogonal design comprises parameterizing the columns of thechannel matrix {tilde over (H)} using the precoders so that the columnsof the channel matrix {tilde over (H)} form two orthogonal vectorsubspaces in which signals from the at least two users are transmitted.6. The method of claim 5 further comprising calculating the parametersused in parameterizing in the precoders using low-complexity algorithmsin which a solution for the parameters is reduced to any one of thesolutions to a set of linear equations characterizing operation of theprecoders, where the solutions of the set of linear equations arenonunique.
 7. The method of claim 6 further comprising selecting theprecoder parameters by selecting one of the solutions to the set oflinear equations characterizing operation of the precoders, whichselected solution generates values for the parameters in the channelmatrix {tilde over (H)} which parameters characterize the precoders witha low demand for computation.
 8. The method of claim 1 where decodingthe signals independently to provide diversity of NM to the at least twousers comprises using maximum-likelihood decoding to separately detectreal and imaginary portions of the signals from the at least two users.9. The method of claim 1 where generating an orthogonal structure ofsignal vectors in corresponding precoders for the at least two users byutilizing a quasi-orthogonal design comprises parameterizing the columnsof the channel matrix {tilde over (H)} using the precoders so that thecolumns of the channel matrix {tilde over (H)} form two orthogonalvector subspaces in which signals from the at least two users aretransmitted; calculating the parameters used in parameterizing in theprecoders using low-complexity algorithms in which a solution for theparameters is reduced to any one of the solutions to a set of linearequations characterizing operation of the precoders, where the solutionsof the set of linear equations are nonunique; selecting the precoderparameters by selecting one of the solutions to the set of linearequations characterizing operation of the precoders, which selectedsolution generates values for the parameters in the channel matrix{tilde over (H)} which parameters characterize the precoders with a lowdemand for computation; and where decoding the signals independently toprovide diversity of NM to the at least two users comprises usingmaximum-likelihood decoding to separately detect real and imaginaryportions of the signals from the at least two users.
 10. An apparatus toachieve full diversity without sacrificing bandwidth and with a linearcomplexity in a wireless system comprising a plurality of orthogonallytransmitting users each having N transmit antennas; at least onereceiver having M receive antennas communicating with the plurality oftransmitting users, which receiver separates the signals using theorthogonality of the transmitted signals, and decodes the signalsindependently to provide diversity of NM to the users; a plurality ofprecoders corresponding to the plurality of transmitting users, whichprecoders are designed using the channel information to cancelinterference among the plurality of signals while achieving a maximumpossible diversity of NM; and where the plurality of transmitting usersorthogonally transmit a plurality of signals using the correspondingplurality of precoders in a plurality of time slots and where theplurality of transmitting users orthogonally transmit a plurality ofsignals utilizing multiple antennas using a corresponding plurality ofprecoders by determining for each time slot of transmission anequivalent channel equation characterized by a channel matrix {tildeover (H)} and a noise vector n for the users, generating an orthogonalstructure of signal vectors in corresponding precoders for the users byutilizing a quasi-orthogonal design comprised of a subspace 1 created bythe first two columns of matrix {tilde over (H)} orthogonal to asubspace 2 created by the second two columns of matrix H.
 11. Theapparatus of claim 10 where the receiver decodes the signalsindependently to provide full diversity to the users achieving a maximumpossible diversity of at least
 4. 12. The apparatus of claim 10 wherethe receiver decodes the signals independently to provide full diversityto the users further comprises achieving full diversity while providinga linear low complexity decoding with any number of users.
 13. Theapparatus of claim 10 where the precoders generating an orthogonalstructure of signal vectors in corresponding precoders for thecorresponding users by utilizing a quasi-orthogonal design comprisesparameterizing the columns of the channel matrix {tilde over (H)} usingthe precoders so that the columns of the channel matrix {tilde over (H)}form two orthogonal vector subspaces in which signals from the users aretransmitted.
 14. The apparatus of claim 13 further comprising precoderswhich determine parameters used in parameterizing the columns of thechannel matrix {tilde over (H)} using low-complexity algorithms in whicha solution for the parameters is reduced to any one of the solutions toa set of linear equations characterizing operation of the precoders,where the solutions of the set of linear equations are nonunique. 15.The apparatus of claim 14 further comprising precoders which select theprecoder parameters by selecting one of the solutions to the set oflinear equations characterizing operation of the precoders, whichselected solution generates values for the parameters in the channelmatrix {tilde over (H)} which parameters characterize the precoders witha low demand for computation.
 16. The apparatus of claim 10 where thereceiver decoding the signals independently to provide full diversity tothe at least two users comprises using maximum-likelihood decoding toseparately detect real and imaginary portions of the signals from the atleast two users.
 17. The apparatus of claim 10 where the precodersgenerating an orthogonal structure of signal vectors in correspondingprecoders for the corresponding users by utilizing a quasi-orthogonaldesign comprises parameterizing the columns of the channel matrix {tildeover (H)} using the precoders so that the columns of the channel matrix{tilde over (H)} form two orthogonal vector subspaces in which signalsfrom the users are transmitted; where the precoders determine parametersused in parameterizing the columns of the channel matrix {tilde over(H)} using low-complexity algorithms in which a solution for theparameters is reduced to any one of the solutions to a set of linearequations characterizing operation of the precoders, where the solutionsof the set of linear equations are nonunique; where the precoders selectthe precoder parameters by selecting one of the solutions to the set oflinear equations characterizing operation of the precoders, whichselected solution generates values for the parameters in the channelmatrix {tilde over (H)} which parameters characterize the precoders witha low demand for computation; and where the receiver decoding thesignals independently to provide full diversity to the at least twousers comprises using maximum-likelihood decoding to separately detectreal and imaginary portions of the signals from the at least two users.18. Signal processing instructions recorded within memory components ofa communication system for controlling a plurality of orthogonallytransmitting users each having N transmit antennas, at least onereceiver having M receive antennas communicating with the plurality oftransmitting users, which receiver separates the signals using theorthogonality of the transmitted signals, and decodes the signalsindependently to provide diversity of NM to the users, a plurality ofprecoders corresponding to the plurality of transmitting users, whichprecoders are designed using the channel information to cancelinterference among the plurality of signals while achieving a diversityof NM, where the plurality of transmitting users orthogonally transmit aplurality of signals using the corresponding plurality of precoders in aplurality of time slots, and where the plurality of transmitting usersorthogonally transmit a plurality of signals utilizing multiple antennasusing a corresponding plurality of precoders by determining for eachtime slot of transmission an equivalent channel equation characterizedby a channel matrix {tilde over (H)} and a noise vector n for the users,generating an orthogonal structure of signal vectors in correspondingprecoders for the users by utilizing a quasi-orthogonal design comprisedof a subspace 1 created by the first two columns of matrix {tilde over(H)} orthogonal to a subspace 2 created by the second two columns ofmatrix {tilde over (H)}.